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**TL;DR Summary:**I approach a rocket acceleration problem using two approaches: F=d(m*v)/dt and F=ma. The resulting differential equations are different. What am I doing wrong?

We have a ship with a mass-reaction rocket engine floating in space.

The initial mass of the ship (including fuel) is m0 [kg].

The rocket produces a constant thrust F [N].

The burn rate of the fuel is R [kg/s].

The intitial speed v of the ship is 0 m/s.

What is the velocity/acceleration of the ship?

I first approached this problem like this:

F = d(m*v)/dt (using the momentum version of Newton's 2nd Law)

F = m dv/dt + v dm/dt

F = (m0 - R t) dv/dt + v(-R)

dv/dt = (F + R v) / (m0 - R t)

OK, so it's a differential equation for v.

Next, I approached it like this:

F = m a (the more commonly encountered version of Newton's 2nd Law)

F = (m0 - R t) a

a = F / (m0 - R t)

but a is also dv/dt, isn't it? so I get

dv/dt = F / (m0 - R t)

Compare this with dv/dt from the first approach. They're different. I'm missing a whole term Rv/(m0-Rt).

What am I doing wrong?

Thanks for reading this far.